Common Knowledge in Email Exchanges Krzysztof R. Apt CWI, Amsterdam, the Netherlands , University of Amsterdam based on joint work with Floor Sietsma, CWI Common Knowledge in Email Exchanges – p. 1/2
History: Common, Knowledge COMMON, something that belongs to all alike, in contradistiction to proper, peculiar. KNOWLEDGE is defined, by Mr Locke, to be the perception of the connection and agreement, or disagreement and repugnancy, or our ideas. Encyclopaedia Britannica By a Society of GENTLEMEN in Scotland Edinburgh MDCCLXXI Common Knowledge in Email Exchanges – p. 2/2
History: Nested Knowledge A. Koestler, Arrow in the Blue , p. 75, 1952. p. 192: “As for the diplomatic informer, he worked, of course for the Deuxième Bureau , and the chief instructed us that we should carefully stick to the fiction that we didn’t know that he knew that we knew it. Common Knowledge in Email Exchanges – p. 3/2
History: Common Knowledge (1) Morris Friedell, On the structure of shared awareness , Working Paper, 1967. p.2: “The final section, Economics, will deal with A thinks B thinks ... reasoning in formal games.” A. Koestler, Arrow in the Blue , 1952. p. 279: “It is common knowledge that if you plant a flag on a hitherto uncharted island or territory, you have staked a claim to that territory on behalf of the nation which the flag represents.” Common Knowledge in Email Exchanges – p. 4/2
ome History: Common Knowledge (2) Common Knowledge in Email Exchanges – p. 5/2
History: Email Ray Tomlinson (BBN): 1971. Ted Meyer and Austin Henderson(BBN): BCC feature added in 1975. Common Knowledge in Email Exchanges – p. 6/2
Story 1: Why BCC is Useful FYI: Wouter Bos knows Trijntje Oosterhuis. Source: Wouter Bos e-mailt per ongeluk zijn netwerk rond . NRC Handelsblad, 7th October 2010. Common Knowledge in Email Exchanges – p. 7/2
Story 2: Forwarding I got an email from my Chinese postdoc Helen with a CC to her husband Bo. Common Knowledge in Email Exchanges – p. 8/2
Story 2: Forwarding I got an email from my Chinese postdoc Helen with a CC to her husband Bo. I forwarded it to Floor. Common Knowledge in Email Exchanges – p. 8/2
Story 2: Forwarding I got an email from my Chinese postdoc Helen with a CC to her husband Bo. I forwarded it to Floor. Floor replied to my forward to Bo and Helen with a BCC to me. Common Knowledge in Email Exchanges – p. 8/2
Story 2: Forwarding I got an email from my Chinese postdoc Helen with a CC to her husband Bo. I forwarded it to Floor. Floor replied to my forward to Bo and Helen with a BCC to me. I forwarded the last email to Helen and Bo with a BCC to Floor. Common Knowledge in Email Exchanges – p. 8/2
Story 2: Forwarding I got an email from my Chinese postdoc Helen with a CC to her husband Bo. I forwarded it to Floor. Floor replied to my forward to Bo and Helen with a BCC to me. I forwarded the last email to Helen and Bo with a BCC to Floor. Do we all have common knowledge of Floor’s reply? Common Knowledge in Email Exchanges – p. 8/2
Messages Agents: finite set Ag , Notes: further unspecified. s ( i, l, G ) ; message containing note l , sent by agent i to group G , f ( i, l.m, G ) ; forwarding by i of message m with added note l , sent to group G , S ( m ) = { sender of m } , R ( m ) = receivers of m . Special cases: reply: f ( i, l.m, G ) , with G = S ( m ) , where i ∈ R ( m ) , reply-all: f ( i, l.m, G ) , with G = S ( m ) ∪ R ( m ) , where i ∈ R ( m ) . Common Knowledge in Email Exchanges – p. 9/2
Emails Email: m B , with m a message, B ⊆ Ag a set of BCC recipients. Examples Passing a note further: s ( i, l, G ) B , s ( j, l, G ′ ) B ′ , where j ∈ G ∪ B , Forwarding an email: m B , f ( i, l.m, G ) C , where i ∈ R ( m ) ∪ B , Forwarding one’s own email: m B , f ( i, l.m, G ) C , where S ( m ) = { i } , BCC reply-all (grrr . . . ): m B , f ( i, l.m, G ) C , where i ∈ B and G = S ( m ) ∪ R ( m ) . Common Knowledge in Email Exchanges – p. 10/2
Modelling Story 2 I got an email from my Chinese postdoc Helen with a CC to her husband Bo. e := m ∅ , where m := s ( H, l, { B, K } ) , Common Knowledge in Email Exchanges – p. 11/2
Modelling Story 2 I got an email from my Chinese postdoc Helen with a CC to her husband Bo. e := m ∅ , where m := s ( H, l, { B, K } ) , I forwarded it to Floor. e ′ := m ′ ∅ , where m ′ := f ( K, m, F ) , Common Knowledge in Email Exchanges – p. 11/2
Modelling Story 2 I got an email from my Chinese postdoc Helen with a CC to her husband Bo. e := m ∅ , where m := s ( H, l, { B, K } ) , I forwarded it to Floor. e ′ := m ′ ∅ , where m ′ := f ( K, m, F ) , Floor replied to my forward to Bo and Helen with a BCC to me. e ′′ := m ′′ { K } , where m ′′ := f ( F, m ′ , { B, H } ) , Common Knowledge in Email Exchanges – p. 11/2
Modelling Story 2 I got an email from my Chinese postdoc Helen with a CC to her husband Bo. e := m ∅ , where m := s ( H, l, { B, K } ) , I forwarded it to Floor. e ′ := m ′ ∅ , where m ′ := f ( K, m, F ) , Floor replied to my forward to Bo and Helen with a BCC to me. e ′′ := m ′′ { K } , where m ′′ := f ( F, m ′ , { B, H } ) , I forwarded the last email to Helen and Bo with a BCC to Floor. f ( K, m ′′ , { H, B } ) { F } . Common Knowledge in Email Exchanges – p. 11/2
Assumptions Each agent has his set of notes, can send/forward his notes, also can send/forward notes he received, also can forward messages he received. Common Knowledge in Email Exchanges – p. 12/2
Some Considerations Are all sets of emails meaningful? Common Knowledge in Email Exchanges – p. 13/2
Some Considerations Are all sets of emails meaningful? No: you can’t forward a note you did not receive. Common Knowledge in Email Exchanges – p. 13/2
Some Considerations Are all sets of emails meaningful? No: you can’t forward a note you did not receive. If one sends/forwards somebody’s else note, then one should have received it. Common Knowledge in Email Exchanges – p. 13/2
Some Considerations Are all sets of emails meaningful? No: you can’t forward a note you did not receive. If one sends/forwards somebody’s else note, then one should have received it. If one forwards a message, then one should have received it. Common Knowledge in Email Exchanges – p. 13/2
States Factual information: FI ( s ( i, l, G )) := { l } , FI ( f ( i, l.m, G )) := FI ( m ) ∪ { l } . State: s = ( E, L ) , where E : set of emails, L = ( L 1 , . . ., L k ) : sets of agents’ notes. Common Knowledge in Email Exchanges – p. 14/2
Legal States Legal state: s = ( E, L ) , such that for some partial ordering ≺ on E for each s ( i, l, G ) B ∈ E , where l �∈ L i , m C ∈ E exists such that m C ≺ s ( i, l, G ) B , i ∈ R ( m ) ∪ C and l ∈ FI ( m ) , Common Knowledge in Email Exchanges – p. 15/2
Legal States Legal state: s = ( E, L ) , such that for some partial ordering ≺ on E for each s ( i, l, G ) B ∈ E , where l �∈ L i , m C ∈ E exists such that m C ≺ s ( i, l, G ) B , i ∈ R ( m ) ∪ C and l ∈ FI ( m ) , for each f ( i, l.m, G ) B ∈ E , where l �∈ L i , some m ′ C ∈ E exists such that m ′ C ≺ f ( i, l.m, G ) B , i ∈ R ( m ′ ) ∪ C and l ∈ FI ( m ′ ) , Common Knowledge in Email Exchanges – p. 16/2
Legal States Legal state: s = ( E, L ) , such that for some partial ordering ≺ on E for each s ( i, l, G ) B ∈ E , where l �∈ L i , m C ∈ E exists such that m C ≺ s ( i, l, G ) B , i ∈ R ( m ) ∪ C and l ∈ FI ( m ) , for each f ( i, l.m, G ) B ∈ E , where l �∈ L i , some m ′ C ∈ E exists such that m ′ C ≺ f ( i, l.m, G ) B , i ∈ R ( m ′ ) ∪ C and l ∈ FI ( m ′ ) , for each f ( i, l.m, G ) B ∈ E some m C ∈ E exists such that m C ≺ f ( i, l.m, G ) B and i ∈ S ( m ) ∪ R ( m ) ∪ C . Common Knowledge in Email Exchanges – p. 17/2
Language to Discuss Emails ϕ ::= m | i ◭ m | ¬ ϕ | ϕ ∧ ϕ Abbreviation: � � m B ::= m ∧ i ◭ m ∧ ¬ i ◭ m i ∈ S ( m ) ∪ R ( m ) ∪ B i �∈ S ( m ) ∪ R ( m ) ∪ B Semantics Take s = ( E, L ) . s | = m ∃ B : m B ∈ E iff s | = i ◭ m ∃ B : m B ∈ E and i ∈ S ( m ) ∪ R ( m ) ∪ B iff s | = ¬ ϕ s �| = ϕ iff s | = ϕ ∧ ψ s | = ϕ and s | = ψ iff Common Knowledge in Email Exchanges – p. 18/2
Epistemic Language Add C A ϕ to the language discussing emails. Intuition: C A ϕ iff group A commonly knows ϕ . Semantics Take s = ( E, L ) . s ′ | = ϕ for any legal s ′ s.t. s ∼ A s ′ . s | = C A ϕ iff ∼ G is the reflexive, transitive closure of � i ∈ A ∼ i . How to define ∼ i ? Common Knowledge in Email Exchanges – p. 19/2
∼ i Relation on Emails m B ∼ i m ′ B ′ iff i ∈ S ( m ) , B = B ′ and m = m ′ , or i ∈ R ( m ) \ S ( m ) and m = m ′ , or i ∈ B ∩ B ′ and m = m ′ . Example If i ∈ B ∩ B ′ , then i cannot distinguish m B from m ′ B ′ . Common Knowledge in Email Exchanges – p. 20/2
∼ i Relation on Legal States ( E, L ) ∼ i ( E ′ , L ′ ) iff L i = L ′ i , ∀ m B ∈ E ( i ∈ S ( m ) ∪ R ( m ) ∪ B → ∃ m B ′ ∈ E ′ m B ∼ i m B ′ ) , ∀ m B ′ ∈ E ′ ( i ∈ S ( m ) ∪ R ( m ) ∪ B ′ → ∃ m B ∈ E m B ∼ i m B ) . Common Knowledge in Email Exchanges – p. 21/2
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